IMO the blackmagic pocket cinema is perfectly fine undisturbed if you're just standing/sitting at a desk, but I was thinking more like you'd pair it with something like ATEM in order to have more control. As for battery life, you're probably going to have it plugged into a dummy battery and then a converter and then into the wall. That's kind of the funny thing with nicer cameras though, they usually require a little bit more interaction because they have so many capabilities.

And for capture cards, yeah the elgato camlink is all you really need- the small form factor is a plus, too.

Sony's interface on their camera is pretty awful, it can be hard to navigate and some things need a ton of button presses just to change simple things. However, this clunky interface comes with the benefit of a staggering amount of features that (in general) work better than its competitors. They just have a really good price:performance ratio, a good form factor, best in class autofocus on the a6400+, and a good selection of compatible lenses. It's just a really solid generalist that isn't missing any important features.

I would still recommend a capture card because I feel like it would reduce the potential for overheating- however, I have not tried using the Sony camera link software yet, so maybe it's fine.

For lenses, it depends on what type of look you want. IMO a lens upgrade is probably the last thing to purchase since you're usually looking at a $300+ price tag. For that amount of money, you could have your lighting down pat which will give you a more immediate bonus in quality than a nicer lens would. A basic 16-50 kit lens will perform fabulously with proper lighting, however it will lack significant background blurring and bokeh.

If you're looking for bokeh and background blurring (as most people are), you want a fast lens with a wider aperture and a longer focal length. However, the more towards that end that you go, the harder it will be to focus on the subject, so you must pair it with good lighting and decent autofocus. For a good idea of what to look for in the mid-low range, a 50mm f1.4 (or f1.8) prime lens should do well.

Also a big tip for desktop setups: https://www.youtube.com/watch?v=7RZeivVQaiM

You can slightly modify a single pole monitor mount and use it as a way to hold cameras, lighting, audio, etc. Here's what I have at home: https://i.imgur.com/ix7xSc1.jpg

You can also purchase those twin VESA mount arms on Amazon for like $20, so you can mount a ton of things for relatively cheap.

I already have a separate boom arm for audio, but you can see there's all sorts of good stuff like a (sort of hidden) camera ball mount and jury-rigged LED panels. I made some reflectors and diffusers in order to make sure I'm not sending tons of light out behind me, which helps exaggerate that nice background blur and makes me stand out. I also have a light behind me over my head, which lights up the hair and provides a nice effect. Something to light up my room could be useful, but I have big windows with a good amount of indirect natural light.

As far as what specific lighting you want- it really just depends. If you're willing to spend a little extra time, I would probably avoid ring lights because you can achieve a better effect with LED panels and creative use of diffusers/reflectors. Ring lights are a good plug-and-play budget solution, though.

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Now for choosing a specific camera- this is a good thing to google since there's a ton of specs and most cameras are pretty similar. IMO you don't need in-body image stabilization, so I would leave out the A6500. The A6100-A6300 are so close in price to the A6400 that I'd prefer the A6400 and say it's a choice between the Sony A6000 and the Sony A6400. The A6000 can be had used in the $250 range and is about 85% as versatile as the A6400, so if you don't care too much about the absolute bleeding edge fastest autofocus and some other luxuries, then the A6000 is a very solid pick. Plus, it gives you a lot more cash to spend on a lens and lighting.

I'd say make a choice between the A6000 and A6400, and then look up comparisons between your choice and its competitors and see which one looks more solid. I don't have much personal experience with brands other than Sony, Canon, and Fujifilm, so most of the time I'm just googling and comparing features, sensors, etc.

Hope this helps!